Exponential Lasso: robust sparse penalization under heavy-tailed noise and outliers with exponential-type loss
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指数 Lasso：重尾噪声与离群点下的稳健稀疏惩罚
- 研究问题
- 摘要称，经典 Lasso 依赖平方损失，在离群点和重尾噪声下模型选择与估计可能不可靠；本文研究如何在高维稀疏惩罚框架内实现兼顾高斯噪声下统计效率与污染稳健性的估计。
- 方法与思路
- 作者提出指数 Lasso，在 Lasso 框架中引入指数型损失函数，使大残差影响平滑下降、小误差近似二次；理论上给出收敛率保证，计算上用 Majorization-Minimization 算法迭代求解一系列加权 Lasso 子问题。
- 材料陈述的贡献
- 摘要称该方法在理想条件下达到与经典 Lasso 匹配的强统计收敛率，同时在重尾污染下保持稳健；数值实验显示其在污染情形优于经典 Lasso，在高斯噪声下也保持较强表现，并提供 R 包 heavylasso。
- 阅读关注点
- 应核对指数损失的具体形式与调参方式、稳健性所需的重尾或污染假设、收敛率的确切条件与速率、MM 算法的收敛性与计算成本，以及数值实验的生成机制和对比基准是否公平。
- 阅读边界
- 仅凭摘要无法判断定理假设是否现实、有限样本表现、调参敏感性、与既有稳健 Lasso 方法的相对优劣；该文为 arXiv 预印本，未经同行评审或期刊接收验证。
原文摘要
In high-dimensional statistics, the Lasso is a cornerstone method for simultaneous variable selection and parameter estimation. However, its reliance on the squared loss function renders it highly sensitive to outliers and heavy-tailed noise, potentially leading to unreliable model selection and biased estimates. To address this limitation, we introduce the Exponential Lasso, a novel robust method that integrates an exponential-type loss function within the Lasso framework. This loss function is designed to achieve a smooth trade-off between statistical efficiency under Gaussian noise and robustness against data contamination. Unlike other methods that cap the influence of large residuals, the exponential loss smoothly redescends, effectively downweighting the impact of extreme outliers while preserving near-quadratic behavior for small errors. We establish theoretical guarantees showing that the Exponential Lasso achieves strong statistical convergence rates, matching the classical Lasso under ideal conditions while maintaining its robustness in the presence of heavy-tailed contamination. Computationally, the estimator is optimized efficiently via a Majorization-Minimization (MM) algorithm that iteratively solves a series of weighted Lasso subproblems. Numerical experiments demonstrate that the proposed method is highly competitive, outperforming the classical Lasso in contaminated settings and maintaining strong performance even under Gaussian noise. Our method is implemented in the \texttt{R} package \texttt{heavylasso} available on Github: https://github.com/tienmt/heavylasso